Solving Equations with Variables on Both Sides
Q1. Solve each equation. Show all working.
(a) 3x = 2x + 7 (b) 5x − 3 = 2x + 6 (c) 4x + 5 = x − 4 (d) 7x − 2 = 3x + 10 (e) 2x + 8 = 5x − 4
Q2. Solve each equation. Check your answer.
(a) 2(x + 3) = x + 10 (b) 3(x − 1) = 2(x + 4) (c) 4(2x + 1) = 3(3x − 2)
(d) 5(x + 2) = 3(x + 6) (e) 2(3x − 1) = 4(x + 3)
Q3. Solve each equation.
(a) x/2 + 3 = x/4 + 5 (b) x/3 − 1 = x/6 + 2 (c) 2x/5 + 1 = x/2 − 1
Q4. Solve each equation.
(a) 3x + 2 = 3x + 5 (b) 2x − 4 = 2(x − 2) (c) 4x + 1 = 4x + 1
Q5. Tom has $(3x + 5) and Sally has $(2x + 12). They have the same amount of money. Write and solve an equation to find x, then find how much each has. 2 marks
Q6. The sum of two consecutive even numbers is 46. If the first number is 2x, write and solve an equation to find the numbers. 2 marks
Q7. A rectangle has length (3x + 2) and width (2x + 5). A square has side length (2x + 4). If the rectangle and square have the same perimeter, find x. 3 marks
Q8. Consider equations of the form ax + b = cx + d.
(a) Solve for x in terms of a, b, c and d. State any restriction. 2 marks
(b) If a = c and b = d, what can you say about the solution? Give an example. 2 marks
(c) If a = c but b ≠ d, what can you say about the solution? Give an example. 2 marks
(d) A student solves 3x + 2 = 5x − 4 by writing 3x − 5x = −4 − 2, giving −2x = −6, so x = 3. Verify this is correct and explain the method used. 2 marks
Answer Key
Q1. (a) x = 7 (b) x = 3 (c) x = −3 (d) x = 3 (e) x = 4
Q2. (a) 2x+6=x+10, x=4 (b) 3x−3=2x+8, x=11 (c) 8x+4=9x−6, x=10 (d) 5x+10=3x+18, x=4 (e) 6x−2=4x+12, x=7
Q3. (a) x/2−x/4=2, x/4=2, x=8 (b) x/3−x/6=3, x/6=3, x=18 (c) 2x/5−x/2=−2, (4x−5x)/10=−2, −x/10=−2, x=20
Q4. (a) 2=5, no solution (b) 2x−4=2x−4, infinitely many solutions (c) 1=1, infinitely many solutions
Q5. 3x+5=2x+12, x=7. Each has $26.
Q6. 2x+(2x+2)=46, 4x+2=46, 4x=44, x=11. Numbers are 22 and 24.
Q7. Rect P = 2(3x+2+2x+5) = 2(5x+7) = 10x+14. Sq P = 4(2x+4) = 8x+16. 10x+14=8x+16, 2x=2, x=1.
Q8. (a) ax−cx = d−b, x(a−c)=d−b, x=(d−b)/(a−c), a≠c (b) If a=c and b=d, 0=0, infinitely many solutions. Example: 2x+3=2x+3. (c) If a=c and b≠d, contradiction (e.g. 0=d−b≠0), no solution. Example: 2x+3=2x+5 gives 3=5. (d) Check: 3(3)+2=11, 5(3)−4=11. Correct. Method: move variable terms to one side and constants to the other by adding/subtracting from both sides.
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